Past exam of the biological physics course of the University of Oxford / 2016 / 1

Treat the membrane as a capacitor, and count the additional positive charge on its inner surface. With radius $r=10^{-6}$ m and length $\ell =10^{-4}$ m, including both end caps givesThe membrane capacitance is $C=C_{m}A=6.3460$ pF. Thus $\Delta Q=C\Delta V=5.3941\times 10^{-13}$ C, corresponding to $\Delta N=\Delta Q/e=3.3667\times 10^6$ monovalent sodium ions. Converting charge to amount of substance using the Faraday constant,Omitting the ends changes the answer to $0.01762$ mM, a one-percent correction. A large membrane potential change therefore requires only about $0.12\%$ of the intracellular sodium population to enter. The bulk liquid remains nearly electrically neutral; excess charge resides chiefly near the cell membrane. This explains why an action potential need not exhaust the concentration difference that drives it, and why fixed reversal potentials are a useful approximation during one spike. This membrane capacitive ion budget assumes sodium supplies the minimum net charging requirement. Simultaneous opposing electric currents can require more sodium entry; repeated spikes eventually require the sodium-potassium ATPase to maintain the gradients.

A typical voltage-gated potassium channel has four subunits, each with six membrane-spanning segments. S1–S4 form a voltage-sensing domain; S5–S6 and their intervening loop form the pore and selectivity filter. Conserved positively charged side chains along S4 move through part of the membrane electric field when the voltage changes. Countercharges and hydrated crevices in the surrounding sensor help make this movement possible. Coupling between the sensor and pore, including the S4–S5 region and S6 gate, changes the probability of opening. A voltage-gated sodium channel usually combines four related domains in one large protein, rather than four identical independent proteins.

The evidence separates charge movement, structural arrangement and functional coupling. A transient ion-channel gating current can persist when permeating ionic electric current is suppressed: some charged parts of the channel itself move. Comparing integrated ion-channel gating charge with channel opening tests their coupling. Charge-neutralization experiments change both total moved charge and voltage dependence, implicating S4 basic residues and countercharges elsewhere; they do not show that S4 alone supplies every charge. ^[ref]

Crystal structure of a mammalian voltage-dependent Shaker family K+ channel by Stephen B. Long and others supplies structural evidence for the arrangement. Voltage-dependent accessibility and optical reports of sensor motion provide complementary functional tests. No one static structure is a movie of gating, and a perturbation that abolishes current can disrupt folding or pore function rather than selectively remove voltage sensing. Agreement between structural, electrical and motion-sensitive measurements supports the mechanism more strongly than any one observation alone.

Write the instantaneous electrical conductance as $g_K(t)={\bar{g}}_{K}n(t{)}^4$. Here ${\bar{g}}_K$ is the maximal electrical conductance of the recorded membrane, $n$ is the activation variable, $V_m$ is inside-minus-outside membrane potential, and $V_K$ is the potassium Nernst equilibrium potential. The factor $V_m-V_K$ is the electrochemical driving voltage; positive $I_K$ conventionally denotes outward positive charge. For an electric current density, ${\bar{g}}_K$ must instead be an electrical conductance per area.

In the Hodgkin-Huxley model, four equivalent independent activation gates must all be permissive, each with probability $n$, so their joint opening probability is $n^4$. At fixed voltage each gate obeysThe model assumes approximately Ohmic open-channel conduction, a well-defined population-average electrical conductance, and an almost constant potassium reversal potential during a brief recording. Four phenomenological gates should not be mistaken for a proof of four physically independent molecular transitions. The simple delayed-rectifier description also omits ion-channel inactivation.

Reducing the available electrical conductance can weaken repolarization, broaden an action potential and alter repetitive firing. Depending on channel class and tissue, excess excitability or impaired recovery can follow; persistent depolarization can also reduce voltage-gated sodium channel availability. Cardiac potassium-current reductions can prolong the action potential and contribute to Long QT syndrome. These are conditional physiological consequences, not a claim that every potassium-current reduction has the same clinical effect.

The average electric current is the number of channels times their ion-channel open probability times the electric current carried by an open channel:Consequently the identifiable answer isTODO (missing input): The source does not supply $V_K$ or the potassium thermodynamic activities for this experiment. A unique numerical electrical conductance cannot follow from the listed electric current, count and ion-channel open probability alone. For the explicitly additional choice $V_K=-90$ mV, ${\gamma }_K=125$ pS. Taking $V_K=-100$ mV instead gives $100$ pS: both match the measured electric current. The table in question 3 describes a skeletal-muscle resting state, and cannot silently supply the conditions of this separate recording.

Rapid permeation is possible because the hydrated pore and coordinating groups of the selectivity filter provide a favorable free-energy landscape for repeated ion transit. Several ions can occupy the filter and their interactions favor onward passage, while the channel remains open for many successive transits. Selectivity need not mean that each transported ion remains tightly trapped for a long time. This is passive downhill transport; it does not consume one ATP molecule per ion crossing.